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Mirrors > Home > MPE Home > Th. List > xmstopn | Structured version Visualization version GIF version |
Description: The topology component of an extended metric space coincides with the topology generated by the metric component. (Contributed by Mario Carneiro, 26-Aug-2015.) |
Ref | Expression |
---|---|
isms.j | ⊢ 𝐽 = (TopOpen‘𝐾) |
isms.x | ⊢ 𝑋 = (Base‘𝐾) |
isms.d | ⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋)) |
Ref | Expression |
---|---|
xmstopn | ⊢ (𝐾 ∈ ∞MetSp → 𝐽 = (MetOpen‘𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isms.j | . . 3 ⊢ 𝐽 = (TopOpen‘𝐾) | |
2 | isms.x | . . 3 ⊢ 𝑋 = (Base‘𝐾) | |
3 | isms.d | . . 3 ⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋)) | |
4 | 1, 2, 3 | isxms 24414 | . 2 ⊢ (𝐾 ∈ ∞MetSp ↔ (𝐾 ∈ TopSp ∧ 𝐽 = (MetOpen‘𝐷))) |
5 | 4 | simprbi 495 | 1 ⊢ (𝐾 ∈ ∞MetSp → 𝐽 = (MetOpen‘𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 × cxp 5676 ↾ cres 5680 ‘cfv 6549 Basecbs 17199 distcds 17261 TopOpenctopn 17422 MetOpencmopn 21303 TopSpctps 22895 ∞MetSpcxms 24284 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-xp 5684 df-res 5690 df-iota 6501 df-fv 6557 df-xms 24287 |
This theorem is referenced by: imasf1oxms 24459 ressxms 24495 prdsxmslem2 24499 tmsxpsmopn 24507 xmsusp 24539 cmetcusp1 25342 minveclem4a 25419 minveclem4 25421 qqhcn 33743 rrhcn 33749 rrexthaus 33759 dya2icoseg2 34049 sitmcl 34122 |
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